Conic Sections

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Conic Sections — Key Facts for JEE Main General second degree: Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0 Discriminant: B² − AC determines type: < 0 → ellipse (circle if also A = C); = 0 → parabola; > 0 → hyperbola Parabola: y² = 4ax; ellipse: x²/a² + y²/b² = 1; hyperbola: x²/a² − y²/b² = 1 Eccentricity: e = distance from point to focus / distance to directrix; e < 1 ellipse, e = 1 parabola, e > 1 hyperbola ⚡ Exam tip: JEE Main tests parametric forms and standard equations — knowing the focus-directrix definition helps in tricky problems!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Conic Sections — JEE Main Study Guide

Parabola:

• Standard: y² = 4ax; focus (a, 0); directrix x = −a; axis: x-axis
• Vertex at origin
• Parametric: (at², 2at)
• Latus rectum: length 4a
• For x² = 4ay: focus (0, a), directrix y = −a

Ellipse:

• Standard: x²/a² + y²/b² = 1 (a > b); focus (±c, 0) where c² = a² − b²; eccentricity e = c/a < 1
• Directrices: x = ±a/e
• Latus rectum: length 2b²/a
• For y²/a² + x²/b² = 1: foci (±c, 0) where c² = a² − b², still on x-axis

Hyperbola:

• Standard: x²/a² − y²/b² = 1; focus (±c, 0) where c² = a² + b²; eccentricity e = c/a > 1
• Directrices: x = ±a/e
• Latus rectum: length 2b²/a
• Rectangular hyperbola: x² − y² = a² or xy = c²; asymptotes are perpendicular (slope ±1 for rectangular)

Parametric forms:

• Parabola y² = 4ax: (at², 2at)
• Ellipse x²/a² + y²/b² = 1: (a cos θ, b sin θ)
• Hyperbola x²/a² − y²/b² = 1: (a sec θ, b tan θ) or (a cosh u, b sinh u)

Tangent equations:

• Parabola y² = 4ax at (at², 2at): ty = x + at²
• Ellipse x²/a² + y²/b² = 1 at (a cos θ, b sin θ): (x cos θ)/a + (y sin θ)/b = 1
• Hyperbola x²/a² − y²/b² = 1 at (a sec θ, b tan θ): (x sec θ)/a − (y tan θ)/b = 1

Condition of tangency:

• Parabola y² = 4ax: line y = mx + c touches if c = a/m

• Ellipse x²/a² + y²/b² = 1: line y = mx + c touches if c² = a²m² + b²

• Hyperbola x²/a² − y²/b² = 1: line y = mx + c touches if c² = a²m² − b²

• Key formula: e = c/a for ellipse/hyperbola; parabola e = 1; focus-directrix: for parabola, any point satisfies PF = PM where M is on directrix

• Common trap: In hyperbola x²/a² − y²/b² = 1, c² = a² + b² (plus, not minus!)

• Exam weight: 1 question per year (4 marks); often combined with circle or in applications

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Conic Sections — Comprehensive JEE Main Notes

Chord of conic: For ellipse/hyperbola, chord with endpoints (x₁, y₁), (x₂, y₂) has equation T = S₁ where midpoint is used For parabola, chord with endpoints (at₁², 2at₁), (at₂², 2at₂): equation is t₁t₂y = 2a(x + at₁t₂)

Director circle:

• Ellipse: x² + y² = a² + b²
• Hyperbola: x² + y² = a² + b² (same formula!) This is the locus of point from which pair of perpendicular tangents can be drawn

Asymptotes of hyperbola: For x²/a² − y²/b² = 1: asymptotes are y = ±(b/a)x (straight lines through centre) For rectangular hyperbola xy = c²: asymptotes are the coordinate axes

Point of intersection of tangents: For parabola y² = 4ax, tangents at (at₁², 2at₁) and (at₂², 2at₂) intersect at point (at₁t₂, a(t₁ + t₂))

Director circle: For ellipse x²/a² + y²/b² = 1: locus of point from which two perpendicular tangents can be drawn is x² + y² = a² + b²

Normal to parabola: At point (at², 2at): equation is y = −tx + 2at + at³ Slope of normal at point is −t

Properties of ellipse:

• Sum of distances from foci to any point on ellipse = 2a (constant)
• Property of reflection: angle of incidence from one focus equals angle of reflection to the other focus

Properties of hyperbola:

• Difference of distances from foci to any point on hyperbola = 2a (constant)
• Asymptotes approach the curve as x, y → ∞

General second degree equation: Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0

• If B² − AC < 0: ellipse (or imaginary ellipse if condition not satisfied)
• If B² − AC = 0: parabola
• If B² − AC > 0: hyperbola (or pair of intersecting lines if det = 0)
• If A = C and B = 0: circle (special ellipse)

Rotation to remove xy term: New axes rotated by angle θ where tan 2θ = 2B/(A − C) After rotation, equation has no xy term; then identify conic from coefficients

Tangents from external point: For ellipse/hyperbola, number of tangents from external point = 2 To find equations, use equation of chord of contact: T = 0 (same as circle)

Normals in ellipse: At (a cos θ, b sin θ): equation is (ax)/cos θ − (by)/sin θ = a² − b²

Conjugate hyperbola: For hyperbola x²/a² − y²/b² = 1, the conjugate hyperbola is −x²/a² + y²/b² = 1 Both have same asymptotes; foci of one are vertices of the other

Rectangular hyperbola: xy = c²; parametric: (ct, c/t) Tangent at (ct, c/t): x/t + y t = 2c Normal at (ct, c/t): t³x − ty = ct⁴ − c/t

Length of latus rectum: Parabola: 4a; Ellipse: 2b²/a; Hyperbola: 2b²/a

Equation reduction: Complete the square to identify centre (if any), axes, and parameters For shifted conic (x − h)²/a² + (y − k)²/b² = 1: centre (h, k), axes parallel to coordinate axes

• Remember: Ellipse: c² = a² − b², e < 1; Hyperbola: c² = a² + b², e > 1; Parabola: e = 1, focus (a,0), directrix x = −a; general conic: discriminant B² − AC determines type
• Previous years: “Find focus and directrix of parabola y² = 8x” [2023]; “If e = 1/2 for ellipse with a = 4, find b” [2024]; “Find equation of tangent to ellipse x²/9 + y²/4 = 1 at point (3 cos θ, 2 sin θ)” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Conic Sections: 1 question per year, 4 marks
• Common patterns: find equation, tangent at point, condition of tangency, focus-directrix, parametric form
• Weight: medium frequency, high scoring

💡 Pro Tips

• For parabola y² = 4ax, parametric form is (at², 2at) — don’t forget the 2 in y-coordinate
• In hyperbola, remember c² = a² + b² (plus, unlike ellipse’s minus)
• For tangency condition, use the specific formula for each conic — they differ
• Director circle is x² + y² = a² + b² for both ellipse and hyperbola — same formula!
• For rectangular hyperbola xy = c², parametric is (ct, c/t) which simplifies many problems

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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